The Symplectic Normal Space of a Cotangent-Lifted Action
Matthew Perlmutter, Miguel Rodriguez-Olmos, M. Esmeralda Sousa-Dias
TL;DR
The paper characterizes the symplectic normal space $N$ for cotangent-lifted Hamiltonian actions on $T^*Q$ by explicitly modeling $N$ in terms of data on the base manifold $Q$ and the group action. It proves that $N$ is (equivariantly) symplectomorphic to $N_oldsymbol extmu oxplus T^*B$, where $N_oldsymbol extmu$ comes from the restricted $H$-action on the coadjoint orbit through $oldsymbol extmu=oldsymbol J(p_x)$ and $B=[ rak h_oldsymbol extmuoldsymbolullet oldsymboleta]^ot_{f S}$; the symplectic form splits as the sum of the KKS form on the orbit and the canonical form on $T^*B$. An explicit $G_{p_x}$-invariant subspace $V$ in $T_{p_x}(T^*Q)$ realizes $N$ (via the $I$-representation), with concrete expressions involving the locked inertia tensor $oldsymbol I(x)$ and a split $ rak g= rak hoxplus rak r$. The authors also derive normal forms in several special cases (totally isotropic momentum, vertical covectors, $ rak h=0$, constant orbit type, and $H riangleleft G_oldsymbol extmu$), facilitating explicit cotangent bundle reduction near singular orbits and informing stability/bifurcation analyses of relative equilibria in mechanical systems.
Abstract
For the cotangent bundle of a smooth Riemannian manifold acted upon by the lift of a smooth and proper action by isometries of a Lie group, we characterize the symplectic normal space at any point. We show that this space splits as the direct sum of the cotangent bundle of a linear space and a symplectic linear space coming from reduction of a coadjoint orbit. This characterization of the symplectic normal space can be expressed solely in terms of the group action on the base manifold and the coadjoint representation. Some relevant particular cases are explored.